2-1 Future value Present value Rates of return Amortization Chapter 2 Time Value of Money

2-2 Will the FV of a lump sum be larger or smaller if we compound more often,? Why? LARGER! If compounding is more frequent than once a year--for example, semiannually, quarterly, or daily--interest is earned on interest more often.

2-3 Semiannually: FV 6 = $100(1.05) 6 = $ % % Annually: FV 3 = $100(1.10) 3 = $

2-4 Rates of return Nominal rate (i Nom ). Periodic rate (i Per ). Effective annual rate

2-5 Nominal rate (i Nom ) Stated in contracts, and quoted by banks and brokers. Not used in calculations or shown on time lines Periods per year (m) must be given. Examples: 8%; Quarterly 8%, Daily interest (365 days)

2-6 Periodic rate (i Per ) i Per = i Nom /m, where m is number of compounding periods per year. m = 4 for quarterly, 12 for monthly, and 360 or 365 for daily compounding. Used in calculations, shown on time lines. Examples: 8% quarterly: iPer = 8%/4 = 2%. 8% daily (365): iPer = 8%/365 = %.

2-7 We will deal with 3 different rates: i Nom = nominal, or stated, or quoted, rate per year. i Per = periodic rate. EAR= EFF% =. effective annual rate

2-8 FV Formula with Different Compounding Periods (e.g., $100 at a 12% nominal rate with semiannual compounding for 5 years) = $100(1.06) 10 = $ FV = PV1.+ i m n Nom mn       FV = $ S 2x5      

2-9 FV of $100 at a 12% nominal rate for 5 years with different compounding FV(Annual)= $100(1.12) 5 = $ FV(Semiannual)= $100(1.06) 10 =$ FV(Quarterly)= $100(1.03) 20 = $ FV(Monthly)= $100(1.01) 60 = $ FV(Daily)= $100(1+(0.12/365)) (5x365) = $

2-10 Effective Annual Rate (EAR = EFF%) The EAR is the annual rate which causes PV to grow to the same FV as under multi-period compounding Example: Invest $1 for one year at 12%, semiannual: FV = PV(1 + i Nom /m) m FV = $1 (1.06) 2 =  EFF% = 12.36%, because $1 invested for one year at 12% semiannual compounding would grow to the same value as $1 invested for one year at 12.36% annual compounding.

2-11 An investment with monthly payments is different from one with quarterly payments. Must put on EFF% basis to compare rates of return. Use EFF% only for comparisons. Banks say “interest paid daily.” Same as compounded daily.

2-12 How do we find EFF% for a nominal rate of 12%, compounded semiannually? EFF% = - 1 ( 1 + ) i Nom m m = ( 1 + ) = (1.06) = = 12.36%.

2-13 EAR (or EFF%) for a Nominal Rate of of 12% EAR Annual = 12%. EAR Q =( /4) 4 - 1= 12.55%. EAR M =( /12) = 12.68%. EAR D(365) =( /365) = 12.75%.

2-14 Can the effective rate ever be equal to the nominal rate? Yes, but only if annual compounding is used, i.e., if m = 1. If m > 1, EFF% will always be greater than the nominal rate.

2-15 When is each rate used? i Nom :Written into contracts, quoted by banks and brokers. Not used in calculations or shown on time lines.

2-16 i Per :Used in calculations, shown on time lines. If i Nom has annual compounding, then i Per = i Nom /1 = i Nom.

2-17 EAR = EFF%: Used to compare returns on investments with different payments per year.

2-18 Amortization Construct an amortization schedule for a $1,000, 10% annual rate loan with 3 equal payments.

2-19 Step 1: Find the required payments. PMT % -1,000

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2-21 Step 2: Find interest charge for Year 1. INT t = Beg bal t (i) INT 1 = $1,000(0.10) = $100. Step 3: Find repayment of principal in Year 1. Repmt = PMT - INT = $ $100 = $

2-22 Step 4: Find ending balance after Year 1. End bal= Beg bal - Repmt = $1,000 - $ = $ Repeat these steps for Years 2 and 3 to complete the amortization table.

2-23 Interest declines. Tax implications. BEGPRINEND YRBALPMTINTPMTBAL 1$1,000$402$100$302$ TOT1, ,000

2-24 $ Interest Level payments. Interest declines because outstanding balance declines. Lender earns 10% on loan outstanding, which is falling. Principal Payments

2-25 Amortization tables are widely used--for home mortgages, auto loans, business loans, retirement plans, and so on. They are very important! Financial calculators (and spreadsheets) are great for setting up amortization tables.

2-26 On January 1 you deposit $100 in an account that pays a nominal interest rate of %, with daily compounding (365 days). How much will you have on October 1, or after 9 months (273 days)? (Days given.)

2-27 i Per = %/365 = % per day. FV=? % -100 Note: % in calculator, decimal in equation.   FV = $ = $ = $

2-28 What’s the value at the end of Year 3 of the following CF stream if the quoted interest rate is 10%, compounded semiannually? % mos. periods 100

2-29 Payments occur annually, but compounding occurs each 6 months. So we can’t use normal annuity valuation techniques.

2-30 1st Method: Compound Each CF % FVA 3 = $100(1.05) 4 + $100(1.05) 2 + $100 = $

2-31 by following these steps: a. Find the EAR for the quoted rate: 2nd Method: Treat as an Annuity EAR = ( 1 + ) - 1 = 10.25%

2-32 b. Use EAR = 10.25% as the annual rate in your calculator: